Storage Valuation
Gas Storage Valuation Methodologies, Quantitative Analysis & Trading Strategies
Presented to the International Centre for Mathematical Sciences in Edinburgh by :
Dario Pavlos Ghazi Tuesday 29 th January 2009 • To obtain the presentation: • Email: [email protected] • Subject: ICMS Presentation
Dario Ghazi Int tel: 252534 Quantitative Analyst Ext tel: +44 (0)1753 492534 Centrica Energy Mob: +44 (0)7789 573957 Fax: +44 (0)1753 431150
Millstream East Maidenhead Road Windsor Berkshire SL4 5GD www.centrica.com Presentation Summary
• Key gas storage functions. • Comparison of different valuation quantitative techniques. • Quantitative analysis of the valuation steps. • Measuring the extrinsic value risk. • Storage trading strategies. • Key gas storage functions. • Comparison of different valuation quantitative techniques. • Quantitative analysis of the valuation steps. • Measuring the extrinsic value risk. • Storage trading strategies. Key gas storage functions Why storage facilities are developed?
• Three key functions behind the development of storage facilities: • Seasonal cycling – Shifting the summer gas production capacity into winter. • Peaking service/peaking deliverability – Responding fast to changing conditions and controlling the peak demand, i.e in cold days. (The advantage of a fast turning storage facility). • Balancing service – Balancing unpredicted small variations in demand/supply. Maintaining quick deliverability when pipeline system has been disrupted. • Key gas storage functions. • Review of different valuation quantitative techniques. • Quantitative analysis of the valuation steps. • Measuring the extrinsic value risk. • Storage trading strategies. Review of different valuation quantitative techniques Hedging with storage using the forward curve
Purchases The Q1-Summer spread provides a good hedge in forward markets.
D J F M A M J J A S O N
Once market monthly contracts trade one can optimise position to fit optimal profile
Initial Hedge Shape to give Q1-Summer p/th Sales Optimal hedge Shape to give 1.057 x (Q1-Summer) p/th Review of different valuation quantitative techniques How easy is it to get the storage value?
• Storage valuation function does not have a closed-form solution, therefore, alternative approaches are suggested. • Storage is probably the most complicated option structure in energy markets. Its complexities exceed even those of power generation. • Storage Value = Withdrawal Revenue – Injection Cost. • However, given the option characteristics that storage gives and the forward curve variability, we need to estimate the solution using simulations for price paths and storage optimal volume spreads. Review of different valuation quantitative techniques Storage Valuation Approaches
• Three main valuation approaches: • Forward optimisation, • Forward dynamic optimisation, • Combinations. Review of different valuation quantitative techniques Forward Optimisation Approach
• Forward optimisation approach:
i , j FV ( F t ; S , I , W , t ) = ∑ ∆ F t V i , j (F t ; S , I , W , t ) i < j Where : FV = Forward Value
F t = Forward Price
V i , j = Optimal Spread Volumes S = Space I = Injection W = Withdrawal
• Value is s.t physical constraints (S, I, W). • Storage value is the summation of all optimal spread volumes multiplied by the corresponding forward prices. • Optimal spread volumes are the net volumes for a given period (i.e a month) spread. Negative spreads are not considered (=0). Review of different valuation quantitative techniques Forward Optimisation Approach continued
• Decision volume variable V is a function of the forward contracts prices at injection and it can be defined as: • The net volumes for a given period (i.e a month) and assuming as an optimal strategy to turn the storage facility’s inventory only once or more, • The net volumes for a given spread period (i.e monthly). • The advantages of using as a decision variable the volumes for a given spread is that this procedure suggests a hedging strategy, i.e long in summer and short in winter. • This hedging strategy is static and is only revised to discount the cash flows. The cash flows will not change regardless of the future evolution of the forward curve. Review of different valuation quantitative techniques Forward Dynamic Optimisation Approach
• Forward Dynamic Optimisation Approach: • This approach takes into account that re-adjustment of the optimal hedges every time the forward curve moves to our favour this will result in a greater storage value without the downside risk. • This strategy is a spread option position.
T
Spread Option Value = FV (Ft ; S , I ,W ,t0 ) + Et [ ∑ max( FV i − FV i−1 ), ]0 0 0 i=t0 +1
• The storage value is a portfolio of complex spread options. • The value of the spread option is a function of the value of the forward contracts plus the risk-neutral expectation of the future forward value spread. Review of different valuation quantitative techniques Forward Dynamic Optimisation Approach continued
• The factor driving the spread option values (storage value) is the variability of the forward curve. • The variability of the forward curve is a function of each single forward contract’s volatility and their correlation structure. • The challenge is to hedge the time value (or else extrinsic value) of the option. Review of different valuation quantitative techniques Decomposing the storage value • Intrinsic Value = Valuation on the Forward Curve. • Rolling Intrinsic = Re-adjusting the Intrinsic value evolution every time the FC moves to our favour. • Extrinsic Value = Total Value – Intrinsic Value. • The three key drivers of the additional value (or time value or extrinsic) are: • The shape of the forward curve. The wider and more variable the summer-winter gap , the greater the additional value. • Volatility, mean reversion, correlation (between forward contracts). Higher volatility, lower MR, lower correlation, higher option time value. • Storage facility’s operational flexibility. Higher flexibility, higher both intrinsic and extrinsic values. Review of different valuation quantitative techniques Decomposing the storage value S10 S40
8.00 16.00 7.00 14.00 6.00 12.00
5.00 10.00
4.00 8.00
3.00 6.00 2.00 4.00 2.00 1.00 0.00 0.00 Intrinsic Rolling Intrinsic Extrinsic Total Value Intrinsic Rolling Intrinsic Extrinsic Total Value
S75 • Intrinsic/Total Value Ratio: 18.00 16.00 Seasonal > Medium > Short Duration 14.00 12.00 10.00 8.00 • Extrinsic/Total Value Ratio: 6.00 4.00 Short Duration > Medium > Seasonal 2.00 0.00 Intrinsic Rolling Intrinsic Extrinsic Total Value Review of different valuation quantitative techniques Decomposing the storage value
Storage Value Split by Service Duration
140.00
120.00
100.00
80.00 Extrinsic 60.00 Intrinsic
40.00 Value (p/th of space) of (p/th Value
20.00
- 10 20 30 40 50 60 70 80 90 100 Service Duration (Days) Review of different valuation quantitative techniques Intrinsic Value Analysis
20 18 16 14 12 10 8 6 4 2 0 S=10 S=40 S=75
Intrinsic (Q-M) Intrinsic (Q-S) Intrinsic (M) Intrinsic (F)
• All four models give roughly the same intrinsic values, therefore, trading on the forward curve without any price variability will generate the same amount of profit regardless of the models or methodology we are using. Review of different valuation quantitative techniques Extrinsic Value Analysis
7 6 5 4 3 2 1 0 S=10 S=40 S=75
Extrinsic (Q-M) Extrinsic (Q-S) Extrinsic (M) Extrinsic (F) • The greater the model flexibility, the more flexible the extrinsic value. • Extrinsic value differs across models due to the different pricing algorithms for gas prices and the numerical methodology in order to solve the storage function optimisation problem. Review of different valuation quantitative techniques How important is extrinsic value? • According to industry experts, the time value (extrinsic) of storage can be from a fraction of the intrinsic (like ¼) up to seven or even eight times the intrinsic value. • The intrinsic and extrinsic values of storage are defined exactly as the intrinsic and extrinsic values of options, since storage is an option. Therefore, extrinsic value has the same properties as in finance: • It increases along with volatility and it declines as we approach maturity (T). The further out we stand, the greater the time -value of storage. • Being able to estimate this time value is to be able to figure out how to unlock extra value, beyond the intrinsic. • The most flexible pricing model is employed by model “M” (stochastic winter-summer spread), therefore, exhibits the most flexible extrinsic value. • Furthermore, some trading strategies are shown at the end on how to generate greater than intrinsic cash flows by using calendar spread options. Review of different valuation quantitative techniques Storage value sensitivity analysis • Storage value is sensitive to both market parameters and physical constraints.
• Market parameters: • Volatility • Correlation • Mean reversion
• Physical constraints: • Injection • Withdrawal • Total space Review of different valuation quantitative techniques Storage value sensitivity to volatility
• Volatility has a positive impact on value. • Vega is one of the Greeks in financial mathematics, it shows the sensitivity of the storage option value to volatility: ∂V/∂σ • Value Sensitivity to Vol: Short duration > Medium duration > Seasonal Storage
Vol Effect On Value Vega Surface
40.0 35.0 20.0 30.0 25.0 15.0 20.0
V a lu e % Change 15.0 10.0 10.0
5.0 5.0 S4 0.0 S3 200 220 240 260 280 300 320 340 360 380 400 dVol 0.0 S2 Vol S=10 S=40 S1 S=75 S=10 S=40 S=75 Storage Review of different valuation quantitative techniques Calendar Spread Option Value Sensitivity to Correlation
CS Option Premium
0.07000000 0.06000000 0.05000000 0.04000000 0.03000000 0.02000000
European0.01000000 Call Price 0.00000000 0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Futures Spread Correlation Review of different valuation quantitative techniques Storage value sensitivity to MR • Energy prices have shown to revert to a long-run level (marginal cost level). • Mean reversion is needed for profitable storage facility in order to be able to cycle gas, however, after a certain point mean-reversion acts as a detrimental factor to storage value. • Mean Reversion has a different kind of impact to seasonal than to short duration storage.
MR effect on Value
350,000,000 300,000,000 250,000,000 200,000,000 150,000,000 100,000,000 50,000,000 - 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
S10 S75
• Seasonal storage should give higher value since it can take greater advantage of expected price swings in a low MR environment. As MR increases, it acts as a detrimental factor especially to the value of the seasonal storage as there are no great price swings expected causing the facility not often filled so much, hence not getting fully exploited. Review of different valuation quantitative techniques Injection Rate Value sensitivity
Intrinsic Value Sensitivity to Injection Rate (S=100, W=10)
27,000,000 26,000,000 25,000,000 24,000,000 23,000,000 22,000,000 Intrinsic Value Intrinsic 21,000,000 20,000,000 0 5 10 15 20 25 30 35 40 45 50 Injection Rate
Total Value Sensitivity to Injection Rate (S=100, W=10)
90,000,000 80,000,000 70,000,000 60,000,000
Value 50,000,000 40,000,000 30,000,000 20,000,000 0 5 10 15 20 25 30 35 40 45 50
Injection Rate Review of different valuation quantitative techniques Withdrawal Rate Value sensitivity
Intrinsic Value Sensitivity to Withdrawal Rate (S=100, I=10)
27,000,000 26,000,000 25,000,000 24,000,000 23,000,000 22,000,000 Intrinsic Value Intrinsic 21,000,000 20,000,000 0 5 10 15 20 25 30 35 40 45 50 Withdrawal Rate
Total Value Sensitivity to Withdrawal Rate (S=100, I=10)
100,000,000 90,000,000 80,000,000 70,000,000 60,000,000 Value 50,000,000 40,000,000 30,000,000 20,000,000 0 10 20 30 40 50 Withdrawal Rate Review of different valuation quantitative techniques Total Space Value sensitivity
Intrinsic Value Sensitivity to Space (I=W=10)
250,000,000.00
200,000,000.00
150,000,000.00
100,000,000.00 Intrinsic Value Intrinsic 50,000,000.00
- 0 2000 4000 6000 8000 10000 Storace Space
Total Value Sensitivity to Space (I=W=10)
350,000,000.00
300,000,000.00
250,000,000.00
200,000,000.00
Value 150,000,000.00
100,000,000.00
50,000,000.00
- 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Storage Space • Key gas storage functions. • Comparison of different valuation quantitative techniques. • Quantitative analysis of the valuation steps. • Measuring the extrinsic value risk. • Storage trading strategies. Quantitative analysis of the valuation steps The key tasks in order to value storage • The storage valuation involves the following stages: • Modelling the gas price evolution process. • Obtain the probability distribution function from thousands of simulated price paths following the pre-specified process. • Determine the optimal gas volume flows according to the observed price spreads given the physical constraints of the storage facility. • Simulate a distribution of the discounted cash flows values by combining the critical probability surface with the optimal volume flows and obtain the average traded profit = withdrawal revenue – injection cost. Quantitative analysis of the valuation steps By modelling the gas prices, can we really guess the future price?
• The reason for modelling the gas spot dynamics is in order to obtain a price distribution that approximates the real distribution. • In financial derivatives valuation (storage valuation) we never try to predict the exact future spot prices but we rather try to get the future price distributional properties right. • Only in that case will our pricing and our hedges be efficient overall. • Price distribution can only be obtained from a simulated price process. Quantitative analysis of the valuation steps Modelling the gas price evolution process • One-factor models capture the random-walk property of prices using a Brownian Motion term. • Two-factor models consider a Brownian Motion process that gravitates around a pre-determined level (long-run marginal cost) using a mean-reverting drift term. • Three -factor models may consider: • Two stochastic mean reverting processes in the drift term of the spot price plus a Brownian Motion, or • A Brownian Motion that mean-reverts plus a Jump process (the Jump size and frequency are calibrated from data), etc. Quantitative analysis of the valuation steps Modelling the gas price evolution process • The pricing algorithm is applied on the given forward curve. • MR Model = Mean-Reverting Model • MRJD = Mean-Reverting Jump Diffusion Model
• Forward Curve for 1 Gas Year (May – April) • Sample price path of a MR (mean-reverting) model • Sample price path of a MRJD (mean-reverting jump diffusion) model Quantitative analysis of the valuation steps Modelling the gas price evolution process
1 Gas Year Forward Curve Sample
0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 £/therm 0.2000 0.1000 0.0000 May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr MR Model MRJD Model
60 60
50 50
40 40
30 30 Price Price 20 20
10 10
0 0 1 2 3 4 5 0 Time 0 1 2 3 4Time 5 6 7 8 910 Storage trading strategies 1 – Factor MR Pricing model, Spot & Forwards trading strategy
Inventory Trajectory
1000000 7
800000 6 5 600000 4 400000 3 SpotPrice
Inventory 200000 2 0
3 8 0 4 0 3 8 0 7 0 3 8 1 0 0 3 8 1 3 0 3 8 1 6 0 3 8 1 9 0 3 8 2 2 0 3 8 2 5 0 3 8 2 8 0 3 8 3 1 0 3 8 3 4 0 3 8 3 7 0 3 8 4 0 0 3 8 4 3 0 3 8 4 6 0 3 8 4 9 0 1 -200000 0 Date
inventory spot price
Storage P&L Histogram Storage Value Split
140 120 100 1,500,000 80 1,000,000 60
Frequency 40 500,000 20
0 - 26,944 680,464 1,333,983 1,987,503 2,641,023 Total Intrinsic Extrinsic P&L Storage trading strategies 1 – Factor MRJD Pricing model, Spot & Forwards trading strategy
Inventory Trajectory
1200000 8
1000000 7 6 800000 5 600000 4 400000 3 Inventory Spot Price 200000 2 0
3 8 04 0 3 8 07 0 3 8 10 0 3 8 13 0 3 8 16 0 3 8 19 0 3 8 22 0 3 8 25 0 3 8 28 0 3 8 31 0 3 8 34 0 3 8 37 0 3 8 40 0 3 8 43 0 3 8 46 0 3 8 49 0 1
-200000 0 Date
inventory spot price
Storage P&L Histogram Storage Value Split
60 1,800,000 1,600,000 50 1,400,000 40 1,200,000 1,000,000 30 800,000 Frequency 20 600,000
10 400,000 200,000 0 -318,686 343,406 1,005,497 1,667,588 2,329,680 2,991,771 3,653,862 - Total Intrinsic Extrinsic P&L Storage trading strategies 1 – Factor B&S Pricing model, Spot & Forwards trading strategy
Inventory Trajectory
1200000 6
1000000 5
800000 4
600000 3 Spot Price Inventory 400000 2
200000 1
0 0 38040 38070 38100 38130 38160 38190 38220 38250 38280 38310 38340 38370 38400 38430 38460 38490
Date
inventory spot price
Storage P&L Histogram Storage Value Split
600 250,000
500 200,000
400 150,000
300 100,000 50,000 Frequency 200 - 100 Total Intrinsic Extrinsic -50,000 0 -13,365,324 -3,247,467 6,870,390 16,988,247 27,106,104 37,223,961 -100,000 P&L -150,000 Quantitative analysis of the valuation steps Price distribution simulation examples
• Simulation example of a mean
reverting model: 0.14
0.12
0.1
0.08 • CASE A: Vol = 300, MR = 10 0.06 Frequency 0.04
0.02
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 50+ Price
0. 4
0. 35 • CASE B: Vol = 100, MR = 30 0. 3 0. 25
0. 2
0. 15
0. 1 Frequency
0. 05
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 50+ Price Quantitative analysis of the valuation steps How appropriate are our modelling techniques? • Key gas storage functions. • Comparison of different valuation quantitative techniques. • Quantitative analysis of the valuation steps. • Measuring the extrinsic value risk. • Storage trading strategies. Measuring the extrinsic value risk Is storage value normal? • The final figure at each valuation is the average full storage value from 1000 or more simulations. • Thus, we can calculate the sample sigma (Standard Deviation) from the simulated final storage values. • However, storage values are non-normally distributed. • Therefore, we cannot apply the symmetric mean -sigma rule to get a 95% value range. Measuring the extrinsic value risk Searching for a risk figure • One possible solution is to: • Directly calculate the 5 th and 95 th percentile (P5 & P95s) of the value from the simulated total values, • Calculate the 95% VaR, • And obtain the minimum extrinsic with a 95% confidence interval. • The same method can be applied for calculating a 99% CI. • Key gas storage functions. • Comparison of different valuation quantitative techniques. • Quantitative analysis of the valuation steps. • Measuring the extrinsic value risk. • Storage trading strategies. Storage trading strategies I Lets trade in the futures spreads, storage will cover us…! • Trading as a simple gas futures spread (intrinsic strategy): • On 1 st Jan, Trader XYZ values a storage lease for seven months: June – December. st st • 1 Jan, the F JUN = 25p/th and F DEC = 50p/th. Thus, on Jan 1 the gas futures spread = 25p/th (intrinsic value of storage). • On 1 st Jan, Trader XYZ sells the Jun-Dec spread (long Jun, short Dec). Meaning he can buy June gas at 25p/th and sell December gas at 50p/th. • On 1 st June, he uses June contract to buy gas@25p/th and puts in storage. • Keeps gas in storage until December and on the1 st of Dec uses Dec contract (F DEC ) to sell gas from storage and lock profit = 25p/th. • For simplicity all costs have been ignored (injection, withdrawal, capacity, financing). Storage trading strategies II Want to make more money? Again storage will cover us…!
• We proved that a storage value is at least equal to the futures spread, however, in reality storage is worth more, and in order to extract this additional value we need to adapt a different strategy. • Trader XYZ sells a Calendar June – December Futures Spread Option on Jan 1 st with expiry on the 1 st of June. • This means that he sells to the option owner the RTB gas from Trader
XYZ using the F JUN contract and the RTS gas to Trader XYZ using the F DEC contract. st • Thus, the CS option has a payoff on the 1 of June: max [ F DEC –FJUN , 0 ] st • Suppose that on Jan 1 : F JUN = 25p/th, F DEC = 50p/th. This means that the intrinsic value of the CS option is 25p/th on Jan 1st . • However, on Jan 1 st the CS option has non-zero time value = 10p/th. • Therefore, on Jan 1 st Trader XYZ sells CS option for 35p/th. st • The CS option value is given by: V CS = (F DEC –FJUN ,0) + @ T = 1 June. Therefore, the option owner will exercise the option if the Jun – December spread widens. • On 1 st Jan, option has non-zero time value (extrinsic). Storage trading strategies II Using storage to back up the Calendar Spread Option trading
st • CASE A: On 1 June, the spread widened: F JUN = 20p/th, F DEC = 60p/th. The option gets exercised from the option’s owner on Trader XYZ . • In order to meet his obligations to the CS option’s owner he has to buy December gas at 60p/th and sell June gas at 20p/th. He has a loss from the option of: -40p/th. • However, the availability of storage allows him to buy June gas at 20p/th and short the F DEC contract (to sell in December at 60p/th). • He stores the June gas, keep it until 1 st December, and sells it using FDEC contract at 60p/th. • From this position Trader XYZ won now 40p/th. • Trader XYZ net position is: -40p/th (CS option loss) + 40p/th (storage and futures spread) + 35p/th (CS option premium) = 35p/th (the premium).
• CASE B: Spread narrowed: F JUN = 30p/th, F DEC = 50p/th. The option is worthless, Trader XYZ keeps the premium. • Again for simplicity all costs have been ignored (injection, withdrawal, capacity, financing). Storage trading strategies II Want to make more money? Again storage will cover us…! • This strategy simply takes advantage of the fact that having storage we can always capture the gas futures spread value (intrinsic strategy) at any point of time that the CS option’s owner decides to exercise and benefit from the current gas future’s spread. • If futures spread moves favourably for the option’s owner then he exercises to get the futures spread benefit and at the same time Trader XYZ can take a position on the same futures spread and using storage he can extract exactly the same value the CS option’s owner got from him. • Therefore, using a storage futures spread position someone can always offset the CS option loss (futures spread). • The CS time value needs to be sufficiently large to cover all costs incurred. • The wider the time gap between the date of selling the CS option (valuation date) and the maturity date (1 st day of the 1 st month on the CS months spread), the greater the time value of the CS option. • This time value can vary from fractions of the futures spread (intrinsic value) to multiples (x8) of the futures spread. • Therefore, the potential option premium, the amount we always end up with when using the CS Option – Storage strategy, can be very significant. • We would expect option premiums to be very big as vol and MR rise to high levels and as correlations breaks down in the future markets and the more time we allow to maturity. Storage trading strategies III Some natural extensions… • Instead of structuring a single CS option, we can design a Basket of CS and get a higher value for it. • In addition to the Basket of CS, we can use straddle strategies for the highly volatile CS and get an ever greater value. • A straddle strategy would be adopted for a highly volatile CS or a Forward contract (F) by going long (buying) simultaneously a Call (RTB) and a Put (RTS) on the CS / F, with the same strike price (K) and maturity (T). • For example, we buy a 1.5p Call and a 1.5p Put on a Dec – Nov CS, 1.5p being the closest K to the current CS price of 1.3p. • The Call premium (0.2p OTM) = 0.075p, and the Put premium (0.2p ITM) = 0.3p. • Our total initial cost for the straddle = 0.375p. • After a few days, the Dec – Nov CS price jumps to 2.2p. • The 1.5p Call is now 0.7p ITM and its premium is now 0.8p. • The 1.5p Put is now 0.7p OTM and its premium is now 0.025p. • If we close the straddle position now we earn 0.825p. • Profit from straddle strategy = 0.825 – 0.375 = 0.50p. Storage trading strategies Locking – in value with alternative strategies
Locked-In Value Using Alternative Strategies
900,000 800,000 700,000 600,000 500,000 400,000 300,000 200,000 100,000 - Intrinsic Strategy BOCS Strategy A BOCS Strategy B BOCS Strategy C
• Intrinsic Strategy = Static forward optimisation on current FC • BOCS A = Trading Forwards, Rolling Basket of CS • BOCS B = Trading Forwards, Rolling Basket of American Style CS • BOCS C = Trading Forwards, Rolling Basket of CS & Straddles • Delta, Gamma & Vega Hedging volumes for BOCS Strategies: C > B > A Questions?